249
Macromolecules 1989,22, 249-256
Micelles Constructed from the Hard Core and the Shell of Symmetrically Arranged Rods. Particle Scattering Function and Mean-Square Radius of Gyration Masukazu Hirata and Yoshisuke Tsunashima* Institute for Chemical Research, Kyoto University, Uji, Kyoto- fu 61 1 Japan. Received April 25, 1988; Revised Manuscript Received June 16, 1988
ABSTRACT: The static light scattering behavior of highly symmetrical “sea urchin” molecules is discussed theoretically. In these molecules the inner and outer parts are constructed from the hard core and the rods of n pieces arranged symmetrically in space, respectively. If the inner part is absent, they show the particle scattering function P(q) and the mean-squareradius of gyration (p) of a regular rodlike star molecule. However, if the inner part is made invisible by using a suitable solvent, P ( q ) shows an oscillatory dampled curve with several peaks which are characteristic of a hollow rodlike star molecule, and the apparent quantity (S2),,, differs from that of a regular rodlike star molecule. Moreover, if the inner and outer parts have different refractive indices from solvents, P(q) varies from one to zero and/or to large values over unity and shows a strange oscillatory curve with several peaks and (p),, changes from plus infinity to minus infinity, which all depend strongly on magnitudes of the weight-fraction refractive index incrementa and the size ratio of the inner and outer parts of the molecules. Introduction Recent developments in the static characterization of well-defined block copolymers in dilute solution by light scattering have made it possible to measure the detailed conformational nature of the intra- and intermolecular phase separation or segregation which occurs as micellar formation when the block copolymers are dissolved in a solvent that is selective for one part of the blocks. Particularly, extensive efforts have been made for neutral block copolymers which are constructed from some chemically different nonionic submolecules. In this case, many static properties of these copolymers in solution result from a van der Waals type of short-range interaction between two segments in the same or different submolecules and from the interaction between the segment and the solvent. Since the possible variety of the molecular structures is determined by the diversity of these interactions that emerges in the polymer-solvent system in question, the varieties that appear in such a neutral block copolymer system are not so splendid as compared with those in an ionic block copolymer system, where the polymers are composed of both ionic and nonionic submolecules. These ionic-nonionic block copolymers are amphiphiles in aqueous solution and show the properties both of neutral block copolymers and of polyelectrolytes. Various novel patterns of conformational structures are expected to be realized in aqueous solution due to the hydrophilic character of the ionic submolecules; in this system, the conformation in solution depends strongly on the degree of ionization and on the interaction between ions fixed on the polymer chain and free ions in the solutions. In the concentration-temperature diagram for such an ionic block copolymer system,l it is well-known that spherical micelles, rodlike micelles, hexagonal liquid crystalline phases, or the hydrated solid are formed, depending on how the solution concentration and temperature are far from the critical micellar concentration (cmc) and temperature. Moreover, the cmc is usually extremely low;2i.e., molecular dispersion of ionic block copolymers in solution is very hard to realize. All of these features are rarely observed for neutral block copolymers in organic ~olvents.~ In salt-free aqueous solutions of ionic-nonionic block copolymers, the electrostatic repulsions between ionized submolecules are very strong. It is thus plausible that the copolymers form a spherical micelle which is composed of a collapsed hydrophobic core with a number of hydrophilic submolecules (rays) extended outward ra-
dially. No conformationalstudy on this type (“sea urchin” like) of micelle has been done so far, though it will give us wider and deeper insight into the intra- and intermolecular characteristics which appear in various forms of micelles. In fact, very recently, we have succeeded in preparation of salt-free aqueous solution of amphiphilic diblock copolymers by synthesizing poly(N-ethyl-2viny1pyridinium)-block-polystyreneand observed the micellar formation of a sea urchin type in high pr~bability.~ We therefore investigate in the present paper the typical static characteristics in solution, i.e., the particle scattering function and the mean-square radius of gyration, of the sea urchin type of molecule. Here the molecule is represented by a model which is constructed from a rigid spherical “core” and an outer “shell” of n spatially symmetrically arranged thin rods. This model is reduced to a rodlike star molecules when the inner core becomes infinitesimmally small. A hollow rodlike star molecules is also realized if the core can be made invisible by choosing a solvent that matches its refractive index.
Particle Scattering Function For rigid molecules, the relative positions between different segments are fixed. Let Rjk be the distance between the jth and the kth segments in a molecule that is oriented to a given direction against the scattering vector q. The particle scattering function for the molecules P(q) is given by N
P ( q ) = N-’C
N
j=1 k = l
(exP(iWRjk))
(1)
with ( ) the configurational average over all the orientation of the molecule and N the total number of segments composed of the molecules. P(q) is normalized to unity at q = 0. Averaged over all the orientation of Rjk to q,the P(q) can be obtained as
As shown in Figure 1,the sea urchin molecule is composed of two parts, an inner rigid core of radius r, and an outer shell of n radially arranged rods of length L. The P(q) can then be expressed by a following summation
250 Hirata and Tsunashima
Macromolecules, Vol. 22, No. 1, 1989
v
2
n-
‘C Figure 1. Model of a sea urchin molecule. The core radius is r, and the lengths of n rods are all L. In this equation, Pi(q),Po(q),and Pio(q)are the partial particle scattering function for the inner core, the outer shell rods, and the interference between the core and the shell, respectively. The quantity (wivi + wove) represents the refractive index increment of the whole molecule with w, and ut the weight fraction and the refractive index increment of the t (t = i and 0)part, respectively. Pi(q)is given by the particle scattering function for the hard sphere6 of radius r,
Pi(q)= ( 3 / ~ ~ ) ~ (xs -i nx cos x ) ~ x = qr, = (5/3)1/2q(S2)i1/2 (sphere)
(4)
where ( S2)iis the mean-square radius of gyration of the sphere. The other particle scattering functions Po(q)and Pio(q)can be obtained in the following way. Particle Scattering Function of the Outer Shell Rods P o ( q ) . Consider that n rods of length L are distributed in space with regularity. Let No be the total number of segments of n rods, and divide No into n subparts of equal number of segments No/n; each rod contains No/n segments. Po(q)can then be expressed by sum of n2partial particle scattering functions Po(q,l,m)as follows N.
N.
n
n
(5)
(0
.Y X’
Figure 2. Spatial geometry on the interference (a) between two rods and (b) between a rod and the core. expression in eq 5, together with eq 6 and 7, is general in form and is applicable to every case where rods are distributed in space with some regularity. Generally, the evaluation of Po(q)is very tedious because Po(q)depends strongly on the relative orientations of a given pair of Ith and mth rods. However, it becomes simpler in the present sea urchin type of molecules since the rods are arranged highly symmetrically. When the molecular center is set in the origin of the coordinate, each rod in the model is arranged on the radial lines directed from the origin to the vertexes of regular polyhedrons or to the face centers of pseudoregular polyhedrons, of which centers are also set on the origin. The one end of each rod is put on an intersection of the radial line with the spherical core surface and another end is set on a vertex of the polyhedron or a face center of the pseudopolyhedron (Figure 2a). Let B(1,m) be the spatial angle between the lth and mth rods; it follows that
[ ( S I ,- s l l ) . ( S m z Here Po(q,l,m)denotes the interference due to all pairs of the segments on the lth and mth rod, e.g., j on the lth rod and k on the mth rod. Rj(l)k(m) represents the magnitude of the distance between this two segments, which are located in the spatial positions RjCl)and Rk(m), respectively. For 1 = m, Pobecomes the intrarod interference and the Po(q,l,l) values are all equal to each other for 1 = 1-n. For I # m, Po is the interrod interference with Po(q,l,m) = Po(q,m,l). Since the lth and mth rods are fixed in space, we can specify the absolute positions of the start and end points for each rod to be Slland SI2for the rod 1 and Sml and Sm2for the rod m. Then we have
+
0)
PI11
(6)
(O,O, rc L 1
=
Ism11 = rc
- SmJI/L2
=
COS
e(l,m)
Po(q,l,m)in eq 6 can then be classified into several subgroups, so that, in each subgroup t , all the angles 6(l,m) should be equal to B,, independent of 1 and m, and that all the Po(q,l,m)sshould give a same value depending only on 6,. If we have d blocks of these subgroups and each subgroup contains c, pieces of equivalent Po(q,l,m),we can rewrite eq 5-7 as Po(q)= n-2[nPo(q,l,0 + 2,CzPo(q,l,m;l 5 for n 2 3. This P(q) feature for RSn molecules is graphically compared in Figure 5 with the P ( q ) for other particular shaped molecules. Here the Kratky plot, i.e., q2Rc2P(q) vs qRc, is employed with RG the root ms radius of gyration of the whole molecule. The molecules examined are rods of length L'(curve R),6 Gaussian coils (curve G),7 regular stars of n Gaussian subchains each having the same radius of gyration RG' (curve GSn),*and spheres (curve S). These P(q)s were calculated by the following equations:
Figure 4. Particle scattering function for rodlike star molecules P(q) with n rods. A broken curve represents n = 2 and the other real curves n = 3-92.
With eq 20-22, we can estimate from eq 15 the values of (S2),,, for the sea urchin molecules of given wt and vt ( t = i and 0).
Results and Discussion In what follows, we first discuss the results on the q dependences of partial particle scattering functions Po(q) and Pio(q)and then on the variation of whole molecular values P ( q ) and ( S 2 )with wt and ut for the sea urchin molecules. P ( q ) for Rodlike Star Molecules. For the star molecules of n rods (abbreviated as RSn), the central hard
P ( q ) = x - l L2'(sin x/x) dz - (sin x / x ) ~ , x = qL'/2 = 3 l I 2 q R ~ (rod) P ( q ) = (2/x2)[exp(-x) - 1 + x], x = (qRc)2 (Gaussian coils)
P(q) = (2/nx)[x - (1- exp(-x)] + ((n- 1 ) / 2 } ( 1e ~ p ( - x ) ) ~ ] , x = n(3n - 2 ) - ' ( q R ~=) ~(qRc')2 (GSn) P ( q ) = see Table 11,
qL = 3lI2qRG (this work) (RSn)
Due to the high intramolecular symmetry, the plotted curves of RSn and S show the characteristic oscillatory
Macromolecules, Vol. 22, No. 1, 1989
Micelles from Symmetrically Arranged Rods 253
Hollow Rodlike Star n= 3 4 6
8
12 20 32 92 (top to bottom)
0
10 qL
20
Figure 7. Particle scattering function due to the interference between the core and the shell P&) for sea urchin molecules with a given size ratio r J L . 0 0
10
20
Figure 6. Particle scattering function for hollow rodlike star molecules Po(q)with n rods. The core to shell size ratio r,/L is taken to be 0.1,0.5,1,and 10. In each subfigure, the eight curves represent Po(q)for n = 3, 4, 6, 8, 12, 20, 32, and 92 from top to bottom.
behavior over the wide range of qRG. Especially at qRG larger than 1.8, RSn curves reveal their distinct characteristics from those of other shapes of molecules. The number n for RSn will be detectable if the P(q) data are obtained at higher qRG, say, more than 4. P ( q ) for Hollow Rodlike Star Molecules. In this case, the molecules are exactly the "shell" of sea urchin molecules where the "core" is made invisible optically. The typical Po(q)curves are shown in Figure 6 as a function of qL for four cases of r,/L = 0.1,0.5, 1,and 5, respectively. In each sub-Figure where r,/L is fixed, the number of rods n is changed as n = 3,4,6,8,12,20,32, and 92, from the top to the bottom. It is found that the oscillatory feature in Po(q)becomes clear as r c / L increases; at a constant n, the Po(q)gives larger peak height and lower peak position (in terms of qL) with increasing r c / L . This feature is responsible to the situations that, with increasing rc, the distance between a pair of rods spreads enough to show some detectable periodicity. Interference Term between the Core and the Shell, Pio(q).Figure 7 shows Pio(q)for sea urchin molecules as a function of qL. As described already, Pio(q)is independent of n and varies with a parameter rc/L. The value of r,/L was changed as 0.001, 0.1,0.2, 0.5,1, 2,5, and 10, as indicated in Figure 7. At rcfL < 0.2, the Pio(q)gives detectable damped oscillatory curves which reach zero asymptotically from the positive region, but at r c / L > 0.5 it decreases quickly to the negative region and then turns back to zero within a comparably small range of qL. The former is a distinct characteristic of the sea urchin molecules with high intramolecular symmetry. Moreover,
Figure 8. Kratky plots for three components of the sea urchin molecules with r,/L = 0.01, 0.1, and 1. Po(-) represents the c w e for the hollow rodlike star part of n = 6, Pi (-- -) for the core part, and Pi, (- - -) for the core-shell interference part. For Pi,the core size rc was expressed in terms of qL with the relation qr, = qL(r,/L).
Figure 7 indicates that the interference effect becomes negligibly small when the core size r, becomes larger than a quarter of the rod length. P ( q ) for Sea Urchin Molecules. Figure 8 shows, as an example, the Kratky plots, (qL)2Pt(q) vs qL, of three components (t = 0,io, and i) of P(q),i.e., Po(!) for hollow rodlike stars of six arms,pio(q)for core-shell interactions, and Pi(q) for cores, respectively. In each case, the molecular size was indicated by the ratio r,/L and it was changed as 0.01, 0.1, and 1as indicated by the figures on the curves. Each curve reveals clearly the oscillatory behavior and shows no plateau constant in larger qL, which is characteristic of the Gaussian chains. Since P(q)for sea urchin molecules can be evaluated from eq 3, the P ( q ) structure depends strongly on wt and ut of the core ( t = i) and the shell ( t = 0)parts of the molecules and as well
Macromolecules, Vol. 22, No. 1, 1989
254 Hirata and Tsunashima r,/L=OOl
r,i
t = 0I
Figure 10. Same plots as in Figure 9 but for r,/L = 0.01, 0.1, 0
10
and 10. The number of rods is 6.
20
6
IO
r,lL-O
3
'/
01
I y . 2
h
0-
v
&3 -I
U
v
0 0
10 qL
20
Figure 9. Kratky plots for sea urchin molecules with n = 6 and r,/L = 1. The weight-fraction refractive index increment of the core party is changed from 20 to -10 as indicated by figures on
the curve.
as on r c / L . The size ratio r,/L depends on wi. For example, if the molecules are diblock copolymers, it follows that
6
-
7
-
P Q
v
r,/L = n1/3[wi1/3/(l - ~ ~ ) ] [ M / ( 4 ~ d ~ / 3 ) ] ~ /=~ / ( M / d , 1) [(nwi)1/3/(1 - W J ] ( R * / L * )(23) where M is the molecular weight of a single block copolymer, di the density of the core, and d, the density of the rod per unit length. R* and L* represent the molar equivalent sphere radius of the core part and the molar length of the shell part of the block copolymer, respectively. When the weight-fraction refractive index increment of the core p a r t y defined by9 y =
WiVi/(WiVi
+ w,v,)
(24)
is introduced, the y becomes a variable changing from +a to -* and is independent of r J L . P ( q ) of eq 3 then becomes a function of y and r,/L alone:
P(q) = Y2Pi(q)+ ( 1 - y)2Po(q)+ 2y(l - Y)Pio(q)
(25)
For eq 25, three special cases can be considered: (1) the core part actually disappears, or solvents are chosen so that (2) the core part should be made invisible or that (3) the total refractive index increment of the polymer should be zero, i.e., wivi + w,v, = 0. The first case is the rodlike star micelles where wi = 0 (y = 0) and r,/L = 0, and P(q) = PO(qlrlo, as shown already in Figure 4. The second case gives hoflow rodlike star micelles where vi = 0 (y = 0) and r,/L # 0, and the P ( q ) is given by P(q) = P,(q),,o and is illustrated already in Figure 6. In the third case where y = fa,no light is scattered a t zero angle but the characteristic scattered light can be observed at finite angles; in the very small qL region, the P(q)continues to increase with qL from 1 to a large positive value in the case y = + m , while it decreases at first and then increases up to a large positive value with a sharp minimum in the case y = -a. Even in other usual cases where -A < y C A ( A = positive
0 qL
10 qL
20
Figure 11. Plots of the particle scattering function P(q) against qL for sea urchin molecules with n = 32. Four cases of r,/L = 0.01, 0.1, 1,and 10 are shown by a pair of subfigures (y 2 0 and y 5 0) from the top left to the bottom right.
value), the P(q) estimated from eq 25 still shows strange and complex behavior of oscillatorycharacter. The strange behavior is caused mainly by the magnitudes of outer shell contribution Po(?)and of cross-term contribution Pi,(q), as demonstrated in Figure 8. The characteristic behavior described here will be show in detail in Figures 9-11. Figure 9 shows an example of the (qL)'P(q) vs qL plots for sea urchin molecules with n = 6 at r,/L = 1. Here, the parameter y was changed from negative to positive values as indicated by the figures on the curves. At y > 1, (qL)2P(q)shows several peaks which decrease both in the heights and in the initial slopes (at qL = 0) with decreasing y. At y < 1, the first peak decreases both in height and in the qL position but the second and other peaks increase in height with decreasing y. Especially the change in (qL)'P(q)at small qL, 0 < qL C 2, is very drastic when y is in the region -1 < y < 1. These will be revealed later in the y dependence of (S2),pp.Similar plots are also shown in Figure 10 with changing r,/L as 0.01, 0.1, and 10. Here n = 6 and y was restricted around -1 C y < 1. A t
Micelles from Symmetrically Arranged Rods 255
Macromolecules, Vol. 22, No. 1, 1989
Table I1 Values of P ( q ) for Rodlike Star Molecules of n Rays as a Function of qL and n n 9L 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00
2 1.OOOoo 0.972 77 0.897 34 0.79021 0.672 40 0.562 67 0.472 68 0.405 55 0.357 75 0.322 82 0.294 89 0.270 51 0.248 66 0.229 58 0.213 51 0.200 12 0.18862 0.178 25 0.16864 0.159 79 0.151 86 0.144 88 0.13865 0.132 94 0.127 56 0.122 49 0.11779 0.11351 0.109 62 0.106 00 0.102 57 0.099 29 0.096 19 0.093 31 0.090 64 0.088 14 0.085 77 0.083 47 0.081 28 0.079 21 0.077 27
3 1.000 00 0.972 66 0.895 66 0.78272 0.652 52 0.524 00 0.412 03 0.324 89 0.263 89 0.224 98 0.201 44 0.186 50 0.175 13 0.164 64 0.154 26 0.144 27 0.135 10 0.12692 0.119 56 0.11274 0.106 33 0.100 39 0.095 14 0.090 74 0.087 16 0.084 20 0.081 55 0.078 94 0.076 24 0.073 48 0.070 78 0.068 24 0.065 92 0.06381 0.061 86 0.060 04 0.058 35 0.056 80 0.055 40 0.054 11 0.052 89
4 Lo00 00 0.972 62 0.895 09 0.780 12 0.645 41 0.509 58 0.388 28 0.291 46 0.222 46 0.17889 0.15467 0.142 53 0.13604 0.13086 0.12487 0.11773 0.10999 0.102 39 0.095 37 0.089 07 0.08341 0.078 32 0.07384 0.070 07 0.067 08 0.064 80 0.063 01 0.061 40 0.059 67 0.057 69 0.055 46 0.053 13 0.050 90 0.048 93 0.047 26 0.045 87 0.044 69 0.043 62 0.042 61 0.041 64 0.040 69
6 1.000 00 0.972 62 0.895 07 0.77991 0.644 38 0.506 34 0.380 58 0.276 68 0.198 55 0.145 42 0.113 39 0.097 15 0.091 14 0.090 44 0.091 15 0.090 70 0.087 86 0.082 60 0.075 72 0.068 38 0.061 61 0.056 05 0.051 88 0.048 96 0.046 98 0.045 63 0.044 64 0.043 80 0.042 93 0.041 87 0.040 54 0.038 92 0.037 09 0.035 19 0.033 38 0.031 81 0.030 58 0.029 73 0.029 20 0.028 90 0.028 67
r,/L 50.1, the (qL)2P(q)behavior becomes simpler than that at r,/L = 1 and it resembles that of hollow rodlike micelles. At r,/L = 10, (qL)2P(q)shows high-frequency oscillatory peaks at qL < 6. This is caused by highly symmetrical arrangement of the rods around the core of large rc. Figure 11shows plots of P(q),instead of (qL)2P(q), against qL for sea urchin molecules of n = 32. The size ratio r,/L is changed as 0.01, 0.1, 1, and 10. In the figure, P(q) is drown separately at y 2 0 and y 5 0. At rc/L I 1, drastic changes of P(q) occur around qL 4 when y varies from ca. 1to negative small values. This indicates that the sign of (S2),,, changes from negative to positive. At r,/L = 10, there are oscillatory damped curves with many peaks around qL < 12 and the initial slopes change from positive to negative at y = 2.2. With eq 20-22 and 24, the apparent ms radius of gyration of sea urchin molecules (S2),, can be expressed in terms of y as (S2),
= Y2(S2)i
+ (1 - YI2(S2), + 2Y(l - Y)(S2)iO = Y ( S 2 ) i + (1 - y)(S2)o (26)
since 2 ( S2)i,= ( S2)i+ ( S2),. Thus it is shown that (S2),,,/L2 = [ ( r , / L ) 2+ ( r c / L )+ 1/31 [ ( 2 / 5 ) ( r , / L ) 2+ ( r c / L )+ 1/31y (27) For the rodlike star molecules, wi = 0 (y = 0), r,/L = 0, and we obtain that (S2),, = (S2)o,r,/L=0 = (1/3)L2. For
8 1.00000 0.972 62 0.89507 0.779 91 0.644 36 0.506 21 0.380 11 0.275 37 0.195 62 0.13981 0.104 10 0.083 54 0.073 37 0.06959 0.069 13 0.069 77 0.069 97 0.068 85 0.06606 0.061 72 0.056 29 0.050 41 0.044 77 0.039 98 0.036 45 0.034 29 0.033 34 0.033 20 0.033 40 0.033 47 0.033 13 0.032 27 0.030 97 0.029 37 0.027 67 0.026 02 0.024 56 0.02336 0.02249 0.021 95 0.02167
12 1.OOO00 0.972 62 0.895 07 0.779 91 0.644 34 0.506 10 0.37973 0.274 33 0.193 25 0.135 27 0.096 51 0.072 29 0.058 40 0.051 56 0.049 34 0.04986 0.051 55 0.05305 0.053 35 0.051 90 0.048 65 0.044 03 0.038 75 0.033 59 0.029 19 0.025 93 0.02391 0.022 97 0.02282 0.023 13 0.02356 0.023 85 0.02383 0.023 38 0.022 47 0.021 17 0.019 64 0.018 08 0.016 70 0.015 66 0.015 01
20 1.OOO00 0.972 62 0.895 07 0.779 91 0.644 34 0.506 10 0.379 72 0.274 32 0.193 22 0.135 17 0.096 22 0.071 61 0.057 00 0.048 99 0.045 08 0.043 45 0.042 70 0.041 85 0.040 26 0.037 76 0.034 53 0.030 97 0.027 57 0.024 75 0.02273 0.021 54 0.021 04 0.02093 0.020 91 0.020 68 0.020 09 0.019 08 0.017 73 0.016 21 0.014 70 0.013 35 0.012 27 0.011 54 0.011 14 0.011 04 0.011 16
32 1.000 00 0.972 62 0.895 07 0.779 91 0.644 34 0.506 10 0.379 72 0.274 31 0.193 20 0.135 12 0.096 09 0.071 31 0.056 38 0.047 85 0.043 19 0.040 59 0.038 75 0.036 82 0.034 36 0.031 32 0.027 96 0.024 68 0.021 84 0.019 72 0.018 38 0.017 76 0.017 65 0.017 81 0.017 97 0.017 89 0.017 42 0.016 52 0.015 27 0.013 82 0.012 33 0.010 98 0.009 87 0.009 05 0.008 54 0.008 30 0.008 29
92 1.000 00 0.972 62 0.895 07 0.779 91 0.644 34 0.506 10 0.379 72 0.274 31 0.193 20 0.135 12 0.096 09 0.071 31 0.056 38 0.047 85 0.043 19 0.040 58 0.038 73 0.036 77 0.034 25 0.031 09 0.027 53 0.023 91 0.020 61 0.017 85 0.015 74 0.014 27 0.013 31 0.012 74 0.012 37 0.01205 0.011 66 0.011 12 0.01044 0.009 65 0.008 84 0.008 10 0.007 49 0.007 05 0.006 78 0.006 67 0.006 67
2 1 lo I
101
\
1
f,/L.Z\
Y
Figure 12. Variation of the reduced apparent mean-squareradius of gyration (S2)app/L2as a function of the weight-fraction refractive index increment of the core y for sea urchin molecules with a given size ratio rJL. An d i e d circle at y = 0 represents
the value for a rodlike star molecule.
hollow rodlike star molecules, y = 0 and (S2),,, = (Sa)o. For the case that wivi + w,~, = 0, it is shown that y = f w and we obtain that (S2),, = +- or --, depending on y = +- or -a. For other cases, the values of (S2), change from positive to negative with increase of y. The change occurs at y = 1 + 9(r,/L)2/[6(rc/L)2+ 15(rc/L) + 51>1. Figure 12 shows this behavior in the form of (S2),,,/L2 vs y with r,/L changing from 0.5 to 20. A circle at y = 0 represents the value for rodlike star molecules. (@),,/L2
Macromolecules 1989,22, 256-261
256
References and Notes is independent of n and the absolute value is larger at larger rJL. The former is in clear contrast to the molecule (1) Mazer, N. A. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum: New York, 1985; Chapter 8. where the outer shell is constructed from the n pieces of (2) For example: Selb, J.; Gallot, Y. In Developments in Block "Gaussian subchains". In this molecule, as will be deCopolymers 2; Goodman, I., Ed.; Elsevier: London, New York, scribed in detail in a forthcoming paper,'" ( S 2 ) a p p / ( R G ' ) 2 1986. depends strongly on n and it shows similar behavior as in (3) Recent reviews of neutral block copolymers have been given by: Burchard, M. Adv. Polym. Sci. 1983, 48, 1. Figure 12 only at n m. (4) Hirata, M.; Nemoto, N.; Tsunashima, Y.; Kurata, M. Proceedings of the 19th Yamada Conference on Ordering and Organization in Ionic Solutions, Kyoto, 1987. Acknowledgment. We thank Dr. K. Kajiwara for (5) Rayleigh, J. W. Proc. R. SOC. London, A 1914, 90,219. valuable discussion in course of the numerical computation (6) Neugebauer, T. Ann. Phys. (N.Y.) 1943,42, 509. of P,(q). (7) Debye, P. Lecture given at the Polytechnic Institute of Brooklyn, Brooklyn, NY, 1944; J. Phys. Colloid Chem. 1947, 51, 18. SupplementaryMaterial Available: Lists of the algorithms (8) (a) Benoit, H. J. Polym. Sci. 1953,11, 507. (b) Burchard, W. (FORTRANprograms) used for calculating PJq) and Pi&);List Macromolecules 1974, 7, 841. 1 is for the former for n = 32, for example, and List 2 is for the (9) Benoit, H.; Froelich, D. In Light Scattering from Polymer latter (12 pages). Ordering information is given on any current Solutions; Huglin, M. B., Ed.; Academic: London, 1972. masthead page. The P J q ) algorithms for other n values are (10) Tsunashima, Y.; Hirata, M.; Kurata, M. Bull. Znst. Chem. Res., Kyoto Univ., in press. available from the authors upon request.
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A Nearly Ideal Mixture of High Polymers C. A. Traskt and C. M. Roland* Chemistry Division, Code 6120, Naval Research Laboratory, Washington, D.C. 20375-5000. Received April 21, 1988; Revised Manuscript Received July 6, 1988 ABSTRACT: Determination of the mixing free energy of blends of cis-1,4-polyisoprene(PIP) and atactic poly(vinylethy1ene)(PVE) was attempted by estimation of the critical molecular weights of the components. Phase separation could not be induced, however, at the highest available molecular weights, indicating a Flory-Huggins interaction parameter for this mixture that is less than 1.7 X lod. This remarkably low magnitude, in the absence of specific interactions, results from the small change in van der Waals energy upon mixing in combination with a close similarity in the liquid structure of the components. The exchange enthalpy of the blend is significantly increased when the PVE has syndiotactic microstructure, resulting in immiscibility with PIP at moderate molecular weights. The glass transition of the PIP/PVE blend was observed to be unusually broad in the presence of a high concentration of PVE. This breadth is unrelated to phase segregation effects but likely reflects differences between components in the free volume necessary for segmental mobility.
Introduction Ideal solutions result from the random mixing of molecules that have the same size and shape and in which the intermolecular forces between pairs of like segments of each type, as well as between unlike segments, are all equivalent. The mixing enthalpy associated with dispersive interactions (arising from correlation between charge fluctuations) favors phase segregation. For macromolecules, in which the combinatory mixing entropy is small, this usually restricts miscibility to only those mixtures in which the components chemically interact. This combinatory entropy is not negligible however and in blends of cis-1,4-polyisoprene and atactic poly(vinylethy1ene) has been shown to be sufficient to effect miscibility as a result of the small enthalpy change accompanying mixing.'s2 The magnitude of the dispersion energy can be described by a series whose leading term corresponds to the well-known London equation3 where ai is the polarizability of the ith molecule or chain unit, separated by r from the j t h unit, and Ii, is approximately equal to the ionization potential of the species. If no alterations of the local liquid structure accompany mixing, the Flory-Huggins interaction parameter, xij, de-
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scribing the enthalpy and noncombinatory entropy contributions to the free energy, can be directly related to the dispersion interaction energy according to4 X "V = (Eij - Eii/2 - E j j / S ) / R T (2)
A negligible change of van der Waals energy upon mixing is primarily a consequence of similarity in the respective polarizabilities of the blend components. The ideal combinatory entropy change, per lattice (or cham unit) volume, is given for a two-component mixture by4 (3) A S = - R ( & / N i In q$ $ j / N , In $j)
+
where $ is volume fraction, R the gas constant, and N the degree of polymerization. By increasing N , this entropy can be diminished to an extent whereby the dispersive interaction energy is sufficient to effect phase separation. The critical point, or minimum on the spinodal curve, defines a value of the Flory-Huggins interaction parameter below which the system is miscible at all concentrations of the components. The composition at the critical point is deduced by equating the third derivative of the free energy with respect to concentration to zero and solving to obtain5 $i* = Njl/z/(NilIz+ N.'/Z) J (4) with the corresponding critical value of the interaction parameter then given by x* = V R / 2 [ VjNJ-'/' ( + ( VjNj)-'/2]2 (5) 0 1989 American Chemical Society
